C. Trigonometric Functions · Sine Functions

Vertical Translations: f(x) = sin x + c

C2.6 C2.7About 65 minutes

Learning goals

I can describe the effect of c on the graph of f(x) = sin x + c.
I can sketch f(x) = sin x + c by moving the five key points.
I can state the axis, maximum, minimum, domain and range of f(x) = sin x + c.
I can find c from a graph or from the maximum and minimum.

Words to know

transformation
A change to a graph, such as a translation, stretch, compression or reflection.

vertical translation
A shift of every point of a graph straight up or down by the same amount.

mapping notation
A rule such as (x, y) → (x, y + c) that shows where each point of a graph moves.

parameter
A number in an equation, such as c, that can take different values and controls a feature of the graph.

Raising the whole wave

Picture two Ferris wheels with the same size and speed. One is built on the ground; the other sits on the roof of a mall, 20 m higher. The seats on both wheels move up and down in exactly the same way. The only difference is that every height on the roof wheel is 20 m greater. Its graph is the same wave, shifted up. That shift is a vertical translation, and it is the first of three transformations of the sine graph you will study.

Remember from Grade 10

For parabolas, the graph of y = x² + k is the graph of y = x² moved up k units (down if k < 0). Every point (x, y) moves to (x, y + k). The same idea works for any function, including sin x.

The role of c in f(x) = sin x + c

For every x, the output of sin x + c is the output of sin x plus c. So every point on y = sin x moves vertically by c:

(x, y) → (x, y + c)

If c > 0, the graph moves up c units.
If c < 0, the graph moves down |c| units.

What changes and what stays the same?

The axis moves from y = 0 to y = c.
The maximum becomes c + 1 and the minimum becomes c − 1.
The range becomes {y ∈ ℝ | c − 1 ≤ y ≤ c + 1}.
The amplitude (1), period (360°) and domain {x ∈ ℝ} do not change, and neither do the x-values of the maximums and minimums.

x	0°	90°	180°	270°	360°
y = sin x	0	1	0	−1	0
y = sin x + 2	2	3	2	1	2

Each output of y = sin x + 2 is 2 more than the matching output of y = sin x.

The dashed curve is y = sin x. The solid curve is y = sin x + 2: every point has moved up 2 units, and the axis is now y = 2.

Worked example

Example 1: Sketch and describe

Sketch y = sin x + 2 for 0° ≤ x ≤ 360°. State its maximum, minimum, axis, amplitude, period, domain and range.

Start with the five key points of y = sin x: (0°, 0), (90°, 1), (180°, 0), (270°, −1), (360°, 0).
Apply (x, y) → (x, y + 2): (0°, 2), (90°, 3), (180°, 2), (270°, 1), (360°, 2). Plot them and join with a smooth wave.
From the new key points: max = 3, min = 1. Axis: y = (3 + 1) ÷ 2 = 2. Amplitude = (3 − 1) ÷ 2 = 1. Period = 360°.
Domain (all angles): {x ∈ ℝ}. Range: {y ∈ ℝ | 1 ≤ y ≤ 3}.

Max 3, min 1, axis y = 2, amplitude 1, period 360°, domain {x ∈ ℝ}, range {y ∈ ℝ | 1 ≤ y ≤ 3}.

Worked example

Example 2: A downward shift

Describe the transformation and give the key points, axis and range of y = sin x − 1.5.

Here c = −1.5, so the graph moves down 1.5 units: (x, y) → (x, y − 1.5).
Key points: (0°, −1.5), (90°, −0.5), (180°, −1.5), (270°, −2.5), (360°, −1.5).
Axis: y = −1.5. Max = −0.5, min = −2.5, so the range is {y ∈ ℝ | −2.5 ≤ y ≤ −0.5}.
Notice the whole graph is now below the x-axis: this function has no zeros.

Down 1.5; axis y = −1.5; range {y ∈ ℝ | −2.5 ≤ y ≤ −0.5}; no x-intercepts.

y = sin x − 1.5 (solid) is y = sin x (dashed) moved down 1.5 units. Its maximum is −0.5, so it never reaches the x-axis.

Worked example

Example 3: Finding c from a graph

A sine curve has the shape of y = sin x (amplitude 1, maximum at 90°), but its maximum value is 4 and its minimum value is 2. Write its equation.

The axis is halfway: y = (4 + 2) ÷ 2 = 3.
The amplitude is (4 − 2) ÷ 2 = 1, the same as y = sin x, so the only change is a vertical translation.
The axis moved from y = 0 to y = 3, so c = 3.

y = sin x + 3

Using technology to investigate

Graph Y1 = sin(X) and then Y2 = sin(X) + C for several values of C, such as 1, 3, −2 and 0.5 (degree mode, window 0 ≤ x ≤ 360, −4 ≤ y ≤ 5). In a graphing app you can often make c a slider and drag it. Record what happens to the axis, max, min, period and amplitude each time. You should find that only the axis, max, min and range change.

Just for fun

What did y = sin x say when it got on the elevator?

"Please take me up c floors, but don't change my shape."

Hands-on activity

Wave in a box

You need: a long strip of paper (about 30 cm), a pencil, a ruler, a second sheet of paper with an x-axis drawn on it, and tape.

On the strip, draw one cycle of y = sin x (use the five key points; let 1 unit = 2 cm, and 90° = 6 cm). Cut along the curve so you have a wave-shaped edge (a grown-up can help with the scissors).
Place the strip on the sheet with its axis on the x-axis and trace the curve. Label it y = sin x.
Slide the strip straight up 3 cm (1.5 units) without turning it. Trace again and label the new curve y = sin x + 1.5.
Slide it down so its axis is 2 units below the x-axis. Trace and label y = sin x − 2.

What to look for: all three curves are the same shape; the peaks stay above 90° and the troughs above 270°; only the heights change. The y = sin x − 2 curve has max −1 and min −3, so it never touches the x-axis.

Your turn

Describe the transformation that maps y = sin x onto y = sin x + 5.

List the five key points of y = sin x − 3 for one cycle starting at 0°.

State the axis, maximum, minimum and range of y = sin x + 0.5.

Which properties of y = sin x do not change in y = sin x + c?

A sine curve with amplitude 1 has a minimum value of 6. Write its equation in the form y = sin x + c.

Does y = sin x + 1 have zeros? If so, find them from 0° to 360°.

More practice: warm-up

y = sin x − 4 moves y = sin x in which direction, and how far?

What is the axis of y = sin x + 6?

What is the maximum of y = sin x − 2?

What is the period of y = sin x + 10?

Write the mapping rule for y = sin x + 4.

More practice: on your own

A Ferris wheel seat's height is modelled by h = sin θ + c (in units of the wheel's radius). The wheel's centre is 1.2 radii above the ground. Write the model and give the lowest height.

Find the range of y = sin x + c in terms of c, and use it to find c if the range is {y ∈ ℝ | −7 ≤ y ≤ −5}.

For which values of c does y = sin x + c have no zeros?

Find the y-intercept of y = sin x − 2.5 and the point where it reaches its maximum between 0° and 360°.

Explain why the zeros of y = sin x + 0.5 are not 0°, 180° and 360°.

Challenge

Find all zeros of y = sin x + 0.5 for 0° ≤ x ≤ 720°.

The graph of y = sin x is translated so that its range is {y ∈ ℝ | 2.4 ≤ y ≤ 4.4}. Write the equation and find the y-value at x = 150°.

Two functions are f(x) = sin x + 3 and g(x) = sin x − 1. Show that f(x) − g(x) is the same for every x, and say what this means about their graphs.

Check your understanding

Quiz

How is the graph of y = sin x − 3 related to y = sin x?

What is the range of y = sin x + 4?

Which property changes when y = sin x is translated vertically?

A sine curve has amplitude 1, maximum 0 and minimum −2. Its equation is

The point (90°, 1) is on y = sin x. Which point is on y = sin x + 2.5?

How many zeros does y = sin x + 3 have?

For parents and teachers

Big idea: The role of c in f(x) = sin x + c (vertical translation) and sketching the transformed graph with its domain and range (C2.6, C2.7).

How to help: Ask your teen to predict the max, min and axis of a function like y = sin x − 4 before graphing it. Then check with a graphing calculator or a free graphing app in degree mode.

Common difficulties:

Moving the graph left or right instead of up or down.
Forgetting that the maximum and minimum change too (they become c + 1 and c − 1).
Assuming the zeros stay at 0°, 180°, 360°. They usually move, and may disappear.